3.11.57 \(\int \frac {1}{x (c+a^2 c x^2) \text {ArcTan}(a x)^{5/2}} \, dx\) [1057]

Optimal. Leaf size=47 \[ -\frac {2}{3 a c x \text {ArcTan}(a x)^{3/2}}-\frac {2 \text {Int}\left (\frac {1}{x^2 \text {ArcTan}(a x)^{3/2}},x\right )}{3 a c} \]

[Out]

-2/3/a/c/x/arctan(a*x)^(3/2)-2/3*Unintegrable(1/x^2/arctan(a*x)^(3/2),x)/a/c

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Rubi [A]
time = 0.06, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]

[Out]

-2/(3*a*c*x*ArcTan[a*x]^(3/2)) - (2*Defer[Int][1/(x^2*ArcTan[a*x]^(3/2)), x])/(3*a*c)

Rubi steps

\begin {align*} \int \frac {1}{x \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^{5/2}} \, dx &=-\frac {2}{3 a c x \tan ^{-1}(a x)^{3/2}}-\frac {2 \int \frac {1}{x^2 \tan ^{-1}(a x)^{3/2}} \, dx}{3 a c}\\ \end {align*}

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Mathematica [A]
time = 0.99, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x \left (c+a^2 c x^2\right ) \text {ArcTan}(a x)^{5/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)),x]

[Out]

Integrate[1/(x*(c + a^2*c*x^2)*ArcTan[a*x]^(5/2)), x]

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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {1}{x \left (a^{2} c \,x^{2}+c \right ) \arctan \left (a x \right )^{\frac {5}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),x)

[Out]

int(1/x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),x)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{a^{2} x^{3} \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )} + x \operatorname {atan}^{\frac {5}{2}}{\left (a x \right )}}\, dx}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a**2*c*x**2+c)/atan(a*x)**(5/2),x)

[Out]

Integral(1/(a**2*x**3*atan(a*x)**(5/2) + x*atan(a*x)**(5/2)), x)/c

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x/(a^2*c*x^2+c)/arctan(a*x)^(5/2),x, algorithm="giac")

[Out]

sage0*x

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{x\,{\mathrm {atan}\left (a\,x\right )}^{5/2}\,\left (c\,a^2\,x^2+c\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)),x)

[Out]

int(1/(x*atan(a*x)^(5/2)*(c + a^2*c*x^2)), x)

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